In stringology, a seed is a substring that covers a superstring of the original string. Its occurrences can extend beyond the original string’s ends, making the seed concept more flexible than an ordinary cover. This article is about the All-Seeds problem in string algorithms—not seeds for planting.
What is a seed of a string?
Let a string be a finite sequence of symbols. A seed is a factor (a contiguous substring) of that string that covers some superstring containing the original string. To cover a string means that occurrences of the factor collectively include every position in it. Because the covered superstring may extend past the original string’s boundaries, the occurrences that provide coverage may overhang the input on the left, the right, or both.
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This boundary allowance is the key distinction: a seed need not cover the original string by itself. Kociumaka and coauthors describe a seed as a relaxed version of a period in which occurrences of the repeating subword may overlap (their paper).
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| Notion | What must be covered or repeated | What it captures |
|---|---|---|
| Period | A block length for which matching repetitions align through the string under the conventional period definition. | Regular repetition within the string. |
| Cover | Occurrences of a factor must cover every position of the string itself. | A factor that tiles the input through overlapping occurrences. |
| Seed | Occurrences of a factor cover a superstring containing the input; coverage can extend beyond the input’s boundaries. | A broader form of repeating structure that can reveal periodicity not expressed by a cover. |
For example, the survey discusses abcabcabca: its cover is abca, while abc, bca, and cab are seeds that expose a period of length three (survey). The example shows why the notions are not interchangeable: the cover describes coverage of the string itself, while seeds can capture a more permissive periodic structure.
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What is the All-Seeds problem?
Given an input word, the All-Seeds problem asks for a representation of every seed of that word. The distinction between listing and representing matters: a string can have a quadratic number of seeds, so explicitly outputting each one would not fit in linear space. Kociumaka, Kubica, Radoszewski, Rytter, and Walen report a linear-time algorithm that produces a linear-size representation for words over an integer alphabet. Their abstract also says that the representation can be used to derive the shortest seed and the number of seeds (A Linear Time Algorithm for Seeds Computation).
This is a theoretical algorithmic result under the paper’s stated alphabet assumptions, not a benchmark or a claim that every implementation runs in linear time on every input representation. A later intuitive formulation likewise describes a seed through occurrences and prefix or suffix overhangs that together cover all text positions (Dagstuhl proceedings paper).
Why use seeds?
- They generalize covers: allowing coverage of a superstring admits factors that cannot cover the input on their own.
- They expose periodic structure: different seeds can show a shorter repeating pattern than the input’s ordinary cover suggests.
- They are useful as a computational family: the all-seeds representation lets algorithms work with a potentially quadratic collection without storing every seed separately.
Scope of “All Seeds”
“All Seeds” can mean different things outside a technical context. Here it refers specifically to the stringology problem of finding seeds of a finite string. The term does not refer to agricultural seed varieties or a video-game item list.
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